Matlab: more plotting functions, anonymous functions, matrix factorisations, and other features. Fix some typos.

matlab.html.markdown

@@ -20,10 +20,17 @@ something
 like
 this %}
 
+% commands can span multiple lines, using '...':
+ a = 1 + 2 + ...
+ + 4
+
+% commands can be passed to the operating system
+!ping google.com
+
 who % Displays all variables in memory
 whos % Displays all variables in memory, with their types
 clear % Erases all your variables from memory
-clear('A') % Erases a aprticualr variable
+clear('A') % Erases a particular variable
 openvar('A') % Open variable in variable editor
 
 clc % Erases the writing on your Command Window
@@ -43,6 +50,7 @@ lookfor command % Searches for a given command
 % Output formatting
 format short % 4 decimals in a floating number
 format long % 15 decimals
+format bank % only two digits after decimal point - for financial calculations
 fprintf 
 
 % Variables & Expressions
@@ -63,7 +71,7 @@ c = exp(a)*sin(pi/2) % c = 7.3891
 3 > 1 || 4 > 1 % OR -> ans = 1
 ~1 % NOT -> ans = 0
 
-% Logicals can be applied to matricies:
+% Logicals can be applied to matrices:
 A > 5
 % for each element, if condition is true, that element is 1 in returned matrix
 A[ A > 5 ] 
@@ -169,9 +177,18 @@ transpose(A) % Transpose the matrix, without taking complex conjugate
 
 
 % Element by Element Arithmetic vs. Matrix Arithmetic 
+% On their own, the arithmetic operators act on whole matrices. When preceded
+% by a period, they act on each element instead. For example:
 A * B % Matrix multiplication
 A .* B % Multiple each element in A by its corresponding element in B
 
+% There are several pairs of functions, where one acts on each element, and 
+% the other (whose name ends in m) acts on the whole matrix.
+exp(A) % exponentiate each element 
+expm(A) % calculate the matrix exponential
+sqrt(A) % take the square root of each element
+sqrtm(A) %  find the matrix whose square is A
+
 
 % Plotting
 x = 0:.10:2*pi; % Creates a vector that starts at 0 and ends at 2*pi with increments of .1
@@ -181,9 +198,24 @@ xlabel('x axis')
 ylabel('y axis')
 title('Plot of y = sin(x)')
 axis([0 2*pi -1 1]) % x range from 0 to 2*pi, y range from -1 to 1
-plot(x,y1,'-',x,y2,'--',x,y3,':'') % For multiple functions on one plot
-grid on % Show grid; turn off with 'grid off'
 
+plot(x,y1,'-',x,y2,'--',x,y3,':'') % For multiple functions on one plot
+legend('Line 1 label', 'Line 2 label') % Label curves with a legend
+
+% Alternative method to plot multiple functions in one plot. 
+% while 'hold' is on, commands add to existing graph rather than replacing it
+plot(x, y)
+hold on
+plot(x, z)
+hold off
+
+loglog(x, y) % A log-log plot
+semilogx(x, y) % A plot with logarithmic x-axis
+semilogy(x, y) % A plot with logarithmic y-axis
+
+fplot (@(x) x^2, [2,5]) % plot the function x^2 from x=2 to x=5
+
+grid on % Show grid; turn off with 'grid off'
 axis square % Makes the current axes region square
 axis equal % Set aspect ratio so data units are the same in every direction
 
@@ -197,11 +229,19 @@ pcolor(A) % Heat-map of matrix: plot as grid of rectangles, coloured by value
 contour(A) % Contour plot of matrix
 mesh(A) % Plot as a mesh surface
 
-h = figure	%C reate new figure object, with handle f
-figure(h) %M akes the figure corresponding to handle h the current figure
+h = figure	% Create new figure object, with handle f
+figure(h) % Makes the figure corresponding to handle h the current figure
+close(h) % close figure with handle h
+close all % close all open figure windows
+close % close current figure window
 
-% Properties can be set and changed through a figure handle
-h = plot(x, y);
+shg % bring an existing graphics window forward, or create new one if needed
+clf clear % clear current figure window, and reset most figure properties
+
+% Properties can be set and changed through a figure handle.
+% You can save a handle to a figure when you create it.
+% The function gcf returns a handle to the current figure 
+h = plot(x, y); % you can save a handle to a figure when you create it
 set(h, 'Color', 'r') 
 % 'y' yellow; 'm' magenta, 'c' cyan, 'r' red, 'g' green, 'b' blue, 'w' white, 'k' black
 set(h, 'LineStyle', '--')
@@ -209,22 +249,38 @@ set(h, 'LineStyle', '--')
 get(h, 'LineStyle')
 
 
+% The function gcs returns a handle to the axes for the current figure
+set(gca, 'XDir', 'reverse'); % reverse the direction of the x-axis
+
+% To creatw a figure that contains several axes in tiled positions, use subplot
+subplot(2,3,1); % select the first position in a 2-by-3 grid of subplots
+plot(x1); title('First Plot') % plot something in this position
+subplot(2,3,2); % select second position in the grid
+plot(x2); title('Second Plot') % plot something there
+
+
+% To use functions or scripts, they must be on your path or current directory
+path % display current path
+addpath /path/to/dir % add to path
+rmpath /path/to/dir % remove from path
+cd /path/to/move/into % change directory
+
+
 % Variables can be saved to .mat files
 save('myFileName.mat') % Save the variables in your Workspace 
 load('myFileName.mat') % Load saved variables into Workspace 
 
-
 % M-file Scripts
 % A script file is an external file that contains a sequence of statements.
 % They let you avoid repeatedly typing the same code in the Command Window
 % Have .m extensions
 
-
 % M-file Functions
 % Like scripts, and have the same .m extension
 % But can accept input arguments and return an output
-% Also, they have their own workspace (ie. different variable scope)
-% double_input.m - .m file name must be same as function name in file
+% Also, they have their own workspace (ie. different variable scope).
+% Function name should match file name (so save this example as double_input.m).
+% 'help double_input.m' returns the comments under line beginning function
 function output = double_input(x) 
 	%double_input(x) returns twice the value of x
 	output = 2*x;
@@ -234,14 +290,26 @@ double_input(6) % ans = 12
 
 % You can also have subfunctions and nested functions.
 % Subfunctions are in the same file as the primary function, and can only be
-% called from within that function. Nested functions are defined within another
+% called by functions in the file. Nested functions are defined within another
 % functions, and have access to both its workspace and their own workspace.
 
+% If you want to create a function without creating a new file you can use an
+% anonymous function. Useful when quickly defining a function to pass to 
+% another function (eg. plot with fplot, evaluate an indefinite integral 
+% with quad, find roots with fzero, or find minimum with fminsearch).
+% Example that returns the square of it's input, assigned to to the handle sqr:
+sqr = @(x) x.^2;
+sqr(10) % ans = 100
+doc function_handle % find out more
 
 % User input
 a = input('Enter the value: ')
 
-% Reading in data
+% Stops execution of file and gives control to the keyboard: user can examine 
+% or change variables. Type 'return' to continue execution, or 'dbquit' to exit
+keyboard
+
+% Reading in data (also xlsread/importdata/imread for excel/CSV/image files)
 fopen(filename) 
 
 % Output
@@ -249,10 +317,10 @@ disp(a) % Print out the value of variable a
 disp('Hello World') % Print out a string
 fprintf % Print to Command Window with more control
 
-% Conditional statements
-if a > 15
+% Conditional statements (the parentheses are optional, but good style)
+if (a > 15)
 	disp('Greater than 15')
-elseif a == 23
+elseif (a == 23)
 	disp('a is 23')
 else
 	disp('neither condition met')
@@ -316,14 +384,18 @@ NaN
 inf
 
 % Solving matrix equations (if no solution, returns a least squares solution)
-x=A\b % Solves Ax=b
-x=B/a % Solves xa=B
+x=A\b % Solves Ax=b. Faster and more numerically accurate than using inv(A)*b.
+x=b/A % Solves xA=b
+inv(A) % calculate the inverse matrix
+pinv(A) % calculate the pseudo-inverse
 
 % Common matrix functions
 zeros(m,n) % m x n matrix of 0's
 ones(m,n) % m x n matrix of 1's
-diag(A) % Extracts the diagonal elements of a matrix 
+diag(A) % Extracts the diagonal elements of a matrix A
+diag(x) % Construct a matrix with diagonal elements listed in x, and zeroes elsewhere 
 eye(m,n) % Indentity matrix
+linspace(x1, x2, n) % Return n equally spaced points, with min x1 and max x2
 inv(A) % Inverse of matrix A
 det(A) % Determinant of A
 eig(A) % Eigenvalues and eigenvectors of A
@@ -340,13 +412,18 @@ dot(A,B) % Returns scalar product of two vectors (must have the same length)
 transpose(A) % Returns the transpose of A
 flipl(A) % Flip matrix left to right
 
+% Alternative forms for matrices
+[L, U, P] = lu(A) % LU decomposition: PA = LU
+[P, D] = eig(A) % eigen-decomposition: AP = PD, P's columns are eigenvectors and D's diagonals are eigenvalues
+[U,S,V] = svd(X) % SVD: XV = US, U and V are unitary matrices, S has non-negative diagonal elements in decreasing order
+
 % Common vector functions
 max     % largest component 
 min     % smallest component 
 length  % length of a vector
 sort    % sort in ascending order 
 sum     % sum of elements 
-prod    % product of elements 
+prod    % product of elements
 mode	% modal value
 median  % median value 
 mean    % mean value